Question

If A and B are square matrices of order 3 such that\[\left| A \right|=-1,\left| B \right|=3\], then the determinant of 3AB is equal to?
a) -9
b) -27
c) -81
d) 81

Answer 1

Hint:In the given question, we have been asked to find the determinant of 3AB and the determinant of A and the determinant of B is given in the question. In order of find the value of determinant of 3AB, we will use the property of determinant and matrices that is if we have any square matrix of order n then \[\left| KA \right|={{K}^{n}}\left| A \right|\], where n is the order of the matrix.

Complete step by step solution:
As we know that,
For any square matrix of order n,
\[\Rightarrow \left| KA \right|={{K}^{n}}\left| A \right|\], where n is the order of the matrix.
We have given that,
\[\Rightarrow \left| A \right|=-1,\left| B \right|=3\]
And the determinant of AB is equal to the product of the determinant of matrix A and the determinant of matrix B.
Such that,
\[\Rightarrow \left| AB \right|=\left| A \right|\times \left| B \right|\]
We have given the matrices of order 3.
Therefore,
\[\Rightarrow \left| KAB \right|={{K}^{n}}\times \left| A \right|\times \left| B \right|\]
Here,
\[\Rightarrow n=3\]As the order of matrix given is 3, it is given in the question
\[\Rightarrow \left| A \right|=-1,\left| B \right|=3\]
So,
\[\Rightarrow \left| 3AB \right|={{3}^{3}}\times \left| A \right|\times \left| B \right|\]
Putting the value of \[\left| A \right|=-1,\left| B \right|=3\], we get
\[\Rightarrow \left| 3AB \right|={{3}^{3}}\times -1\times 3\]
Simplifying the above, we get
\[\Rightarrow \left| 3AB \right|=27\times -1\times 3\]
Multiplying all the number, we get
\[\Rightarrow \left| 3AB \right|=-81\]
Therefore, the determinant of 3AB is equals to -81.
Hence, the option (c ) is the correct answer.

Note: In order to solve these types of question, students should always remember that use the property of determinant and matrices that is if we have any square matrix of order n then \[\left| KA \right|={{K}^{n}}\left| A \right|\], where n is the order of the matrix. As it may have seen many times that a lot of students usually forgot this property of determinant and matrices and then they are not able to solve the question.